
Here, \(|n\rangle \)  is an eigenstate of \(H_{\rm 2D}\)  with energy \(E_n\) , and \(E_\text{gap}\)  is
the energy at the center of the gap. For a sufficiently large system size, such that the gaps are open for any \(\phi \)  and \(\theta \) , \(P_r(\phi ,\theta )\)  is a well-defined quantity, see Fig. REF (b).
Importantly, this topological characterization manifests as a quantized bulk response of the system, namely, quantized Hall conductance [1]}, [2]}, [3]}, [4]}. Furthermore, the quantized bulk response also implies the appearance of chiral edge modes that cross the spectral gap and are localized at the boundary of the system, known as the bulk-boundary correspondence, see Fig. REF (c) [5]}, [6]}, [7]}, [8]}, [9]}, [10]}.
